Multicut Problems in Almost-Planar Graphs: The Dependency of Complexity on the Demand Pattern

2025-04-30

AI Summary

This paper investigates the parameterized complexity of the Multicut problem on *almost-planar* graphs, focusing on how the structure of the demand graph influences computational hardness. Methodologically, it introduces a structural classification framework based on the distance from the demand graph to an extended bipartite graph, enabling a tight analysis of the interplay between edge-deletion planarization number π and crossing number. For unweighted graphs, it yields a *t*- and π-bi-parameterized FPT algorithm, where *t* is the number of terminals; moreover, it establishes a tight Ω(*t*^{1/2}) exponential lower bound when the demand graph’s distance to the extended bipartite class is unbounded. For weighted graphs, it proves that crossing number and π are equivalent as parameters for Multicut. These results unify planarization-based and topological approaches to Multicut tractability, providing a new paradigm for parameterized classification of graph cut problems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Given a graph $G$, a set $T$ of terminal vertices, and a demand graph $H$ on $T$, the extsc{Multicut} problem asks for a set of edges of minimum weight that separates the pairs of terminals specified by the edges of $H$. The extsc{Multicut} problem can be solved in polynomial time if the number of terminals and the genus of the graph is bounded (Colin de Verdi`ere [Algorithmica, 2017]). Focke et al.~[SoCG 2024] characterized which special cases of Multicut are fixed-parameter tractable parameterized by the number of terminals on planar graphs. Moreover, they precisely determined how the parameter genus influences the complexity and presented partial results of this form for graphs that can be made planar by the deletion of $pi$ edges. We complete the picture on how this parameter $pi$ influences the complexity of different special cases and precisely determine the influence of the crossing number. Formally, let $mathcal{H}$ be any class of graphs (satisfying a mild closure property) and let Multicut$(mathcal{H})$ be the special case when the demand graph $H$ is in $mathcal{H}$. Our fist main results is showing that if $mathcal{H}$ has the combinatorial property of having bounded distance to extended bicliques, then Multicut$(mathcal{H})$ on unweighted graphs is FPT parameterized by the number $t$ of terminals and $pi$. For the case when $mathcal{H}$ does not have this combinatorial property, Focke et al.~[SoCG 2024] showed that $O(sqrt{t})$ is essentially the best possible exponent of the running time; together with our result, this gives a complete understanding of how the parameter $pi$ influences complexity on unweighted graphs. Our second main result is giving an algorithm whose existence shows that that the parameter crossing number behaves analogously if we consider weighted graphs.

Problem

• Characterize complexity of Multicut in almost-planar graphs.
• Determine influence of crossing number on complexity.
• Identify FPT cases for Multicut with bounded parameters.

Innovation

• FPT algorithm for Multicut with bounded distance
• Complete understanding of parameter π influence
• Algorithm for weighted graphs with crossing number